准连续近似在失稳附近的准确性
Accuracy of Quasicontinuum Approximations Near Instabilities
AI总结：
本文通过一维模型问题，使用严格分析、渐近方法和数值实验，研究了基本保守准连续近似在晶格失稳附近的稳定性，发现一致准非局部QC近似正确再现了原子系统的稳定性，而非一致能量基QC近似错误地预测了失稳。
摘要：
晶格缺陷（如裂纹、位错或晶界）的形成和运动发生在晶格构型失去稳定性时，即晶格能量泛函的Hessian矩阵的特征值变为负值时。当原子能量被耦合原子和连续模型的混合能量近似时，近似精度只能保证在原子能量和混合能量都稳定的变形附近。因此，我们提出，为了评估原子到连续耦合方法在失稳附近的预测能力，至少对一些代表性模型问题进行理论分析是必要的，以确定混合能量是否在原子能量失稳开始之前保持稳定。我们建立了一个具有最近邻和次近邻相互作用的一维模型问题，并使用严格分析、渐近方法和数值实验来获得基本保守准连续（QC）近似的这种尖锐稳定性估计。我们的结果表明，一致准非局部QC近似正确再现了原子系统的稳定性，而非一致能量基QC近似错误地预测了失稳，其施加的载荷显著降低，我们通过原子势导数的解析准则描述了这一现象。
英文摘要：
The formation and motion of lattice defects such as cracks, dislocations, or grain boundaries, occurs when the lattice configuration loses stability, that is, when an eigenvalue of the Hessian of the lattice energy functional becomes negative. When the atomistic energy is approximated by a hybrid energy that couples atomistic and continuum models, the accuracy of the approximation can only be guaranteed near deformations where both the atomistic energy as well as the hybrid energy are stable. We propose, therefore, that it is essential for the evaluation of the predictive capability of atomistic-to-continuum coupling methods near instabilities that a theoretical analysis be performed, at least for some representative model problems, that determines whether the hybrid energies remain stable {\em up to the onset of instability of the atomistic energy}. We formulate a one-dimensional model problem with nearest and next-nearest neighbor interactions and use rigorous analysis, asymptotic methods, and numerical experiments to obtain such sharp stability estimates for the basic conservative quasicontinuum (QC) approximations. Our results show that the consistent quasi-nonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energy-based QC approximation incorrectly predicts instability at a significantly reduced applied load that we describe by an analytic criterion in terms of the derivatives of the atomistic potential.
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